topo chemistry nano chemistry lecture 01

7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01

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Dr. Ottorino ORIDr. Ottorino ORIActinium Chemical ResearchActinium Chemical Research

RomeRome

[email protected]@gmail.com

WEST UNIVERSITY OF TIMIWEST UNIVERSITY OF TIMIOARAOARAFACULTY OF CHEMISTRY, BIOLOGY, GEOGRAPHY

DEPARTMENT OF BIOLOGYDEPARTMENT OF BIOLOGY--CHEMISTRYCHEMISTRY

LAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICALLAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICAL--CHEMISTRY FORCHEMISTRY FORNANOSCIENCES & QSARNANOSCIENCES & QSAR(CFSCNQ)(CFSCNQ)

Prof. dr. dr. habil. Mihai V. PUTZProf. dr. dr. habil. Mihai V. PUTZ

Director CFSCNQDirector CFSCNQ

[email protected][email protected]


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Chemical Graph Theory is an important branch of

Theoretical Chemistry.

Its basic element is the so-called molecular

graph G. Vertices V(G) of this graph denote

molecular atoms and edges E(G) are the bonds of

the molecule.

Normally, G corresponds the carbon atomskeleton of the molecule. This is the hydrogen-

depleted representation of the molecular

graphs. Atomic valence is the vertex degree r.


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Example Lets consider a connected graph G

like this one with n nodes and m edges:

n =10; m =11

Chemically, G represents Naphtalene molecule

with n carbon atoms and m bonds.

Vertex degree r=2 or 3 corresponds to atomic

valence. For carbon, silicon compounds r 4.


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Graph Theory properties are transferred to

molecular structures like the following TheFirst Theorem of Graph Theory:

Any graph contains an even number ofvertices of odd degree r

which becomes: Every molecule contains aneven number of atoms of odd valence

Then C61 fullerene and others with odd

carbon atoms are forbidden also by topology.


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In Chemical Graph Theory gr aph invar i ant s or

t opologi cal i ndices are used to correlate theproperties of molecules to their chemical

structure.

A t opologi cal i ndex is defined by a map from a

graph G into a number or a finite string ofnumbers:

Topologica l indicesshould be graph invar i ant,

i.e. isomorphic graphs lead to the same

topological index.


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Isomor phic gr aphs


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Isomor phic gr aphs

EXERCISE are these graphs isomorphic ?

See: Lecture Notes on GRAPH THEORY

Tero Harju (2007), Department of Mathematics

University of Turku, FIN-20014 Turku, Finland

e-mail: [email protected]


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Isomor phic gr aphs


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An impor t ant t opological gr aph invar i ant t he Gr aph Ener gy

See : Ivan Gutman, Bo Zhou and Boris Furtula

The Laplacianenergy Like Invariant is an Energy Like Invariant

MATCH Commun. Math. Comput. Chem. 64 (2010) 85-96

ISSN 0340 - 6253 AND RELATED REFERENCES


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Pr opert i es of t he Gr aph Ener gy


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Then the total -electron energy of fullerenes and nanotubes

is determined solely by their carbon atom content and depends

very little on other structural features.

Pr oper t i es of t he Gr aph Ener gy f or r egular gr aphs,

i .e. whit vert ices wi t h t he same number of bonds (degr ee) r


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Given two vertices u and v in V(G), we

denote by d(u,v) or du,v the distancebetween u and v, i.e. the length of the

shortest path connecting u and vby traveling

along the graph edges (or molecular bonds).

This distance belongs to the invariants of the

graphs conveying crucial information about

the structure.

TECHNICALLY AND COMPUTATIONALLY THIS

IS ONE OF SPECIALTIES OF Prof.PUTZs LAB


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1 5 9 7 3

2 6 10 8 4

After an arbitrarilyarbitrarily numbering of the nodes

we calculate the length dij of the shortestpath connecting pair wise the n atoms.

n=10; m=11

Some cases: d11=0 d12=d21=1

d16=d19=2 .. Etc..

dmax=d14=5


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composing - with all dij entries - the Matrix

D of the chemical distances of G:

1 5 9 7 3

2 6 10 8 4

D 1 2 3 4 5 6 7 8 9 10

1 0 1 4 5 1 2 3 4 2 3

2 1 0 5 4 2 1 4 3 3 2

3 4 5 0 1 3 4 1 2 2 3

4 5 4 1 0 4 3 2 1 3 2

5 1 2 3 4 0 3 2 3 1 2

6 2 1 4 3 3 0 3 2 2 1

7 3 4 1 2 2 3 0 3 1 28 4 3 2 1 3 2 3 0 2 1

9 2 3 2 3 1 2 1 2 0 1

10 3 2 3 2 2 1 2 1 1 0


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D contains interesting structural information.

Lets zoom in it first by sorting its rows:1 5 9 7 3

2 6 10 8 4

D wi

1 0 1 1 2 2 3 3 4 4 5 13

2 0 1 1 2 2 3 3 4 4 5 13

3 0 1 1 2 2 3 3 4 4 5 134 0 1 1 2 2 3 3 4 4 5 13

5 0 1 1 2 2 2 3 3 3 4 11

6 0 1 1 2 2 2 3 3 3 4 11

7 0 1 1 2 2 2 3 3 3 4 118 0 1 1 2 2 2 3 3 3 4 11

9 0 1 1 1 2 2 2 2 3 3 8,5

10 0 1 1 1 2 2 2 2 3 3 8,5


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the action of D in possibly grouping

structurally-equivalent atoms emerges:

1 5 9 7 3

2 6 10 8 4

D wi

1 0 1 1 2 2 3 3 4 4 5 13

2 0 1 1 2 2 3 3 4 4 5 13

3 0 1 1 2 2 3 3 4 4 5 134 0 1 1 2 2 3 3 4 4 5 13

5 0 1 1 2 2 2 3 3 3 4 11

6 0 1 1 2 2 2 3 3 3 4 11

7 0 1 1 2 2 2 3 3 3 4 118 0 1 1 2 2 2 3 3 3 4 11

9 0 1 1 1 2 2 2 2 3 3 8,5

10 0 1 1 1 2 2 2 2 3 3 8,5


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Row half-sum wi invariant provides atom-i

contribution to Wiener index Wof G:

W = ij dijW = wiw1 = 13, w5 = 11

W = 109 w9= 8,5We call wi the Wiener weight of atom-i; lets

zoom again in D by sorting by wi ..

1 5 9 7 3

2 6 10 8 4


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the action of wi in suggesting possible

most-stable atoms emerges:

1 5 9 7 3

2 6 10 8 4

D wi

9 0 1 1 1 2 2 2 2 3 3 8,5

10 0 1 1 1 2 2 2 2 3 3 8,5

5 0 1 1 2 2 2 3 3 3 4 11

6 0 1 1 2 2 2 3 3 3 4 11

7 0 1 1 2 2 2 3 3 3 4 11

8 0 1 1 2 2 2 3 3 3 4 11

1 0 1 1 2 2 3 3 4 4 5 13

2 0 1 1 2 2 3 3 4 4 5 13

3 0 1 1 2 2 3 3 4 4 5 13

4 0 1 1 2 2 3 3 4 4 5 13


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Exercise (HOMEWORK): for any (or all) of the

below PAH structures:

Write the ADJACENCY MATRIX

Write the DISTANCE (BONDING) MATRIX

Calculate the Wiener indexby making use of molecular symmetry


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Much attention is devoted on FULLERENES.

Luckily they admit a representation withplanar molecular graphs the Schlegel

d iagramBuckminsterfullerene C60


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Planar connected graphs fulfil Eulers theorem

C20nv nf ne 2

nv=20;ne=20*3/2=30

nf=12whereisthe12th

face?20+1230=2W=500=20*w1


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Wiener index comutation of C20 fullerene

C20 W=500=20*w1w1=25

Shells Atoms k s*k1 3 3

2 6 12

3 6 18

4 3 12

5 1 550

wi 25W=n*wi 500


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Ri-prediction of

the magic LEM:

C28H4


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Topological prediction of saturn like-fullerane C50H10 from C50D5h 10 equatorial reactive nodes (maximal wi ):

NOT BAD AT ALL FOR A PURE TOPOLOGICAL WIENER BASEDNOT BAD AT ALL FOR A PURE TOPOLOGICAL WIENER BASED

CALCULATIONS OF A MILLISECOND!CALCULATIONS OF A MILLISECOND!

Ri-prediction of

the SATURN:

C50H10


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Lets focus on the minimal vertices in green givingthe minimal contribution w to wiener index W

w=8.5

Minimal vertices are distinguished atoms in a graph,they are in fact the most compactly embedded in the

structure.

Now a question naturally arises: how to measurethe ability of the remaining atoms in contributing tothe overall compactness of graph G?

1 5 9 7 3

2 6 10 8 4


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A possible solution consists in computing thefollowing invariant :

= W/Nw 1

= 1.282

Index , called topological efficiency indextopological efficiency index, firstappeared in Ori Ottorino, Cataldo Franco and GraovacAnte (2009) Topological Ranking of C28 FullerenesReactivity, Fullerenes, Nanotubes and CarbonNanostructures, 17:3,308 323.

1 5 9 7 3

2 6 10 8 4


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It is very interesting noticing that for the the

buckminsterfullerene C60 Ih we have:

= 1 C60 is in fact a very round hollow cage showing,

for any of the N=60 nodes, the same {bim}={3 6 8

10 10 10 8 3 1} coordination shells with M=9;wi=139=w and therefore W=N*w=8340. In this

stable structure any node works as minimalminimal

nodenode.


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Assuming as a measure of sphericity it gives

interesting (and sometime unexpected) results: Benzene is spherical =1;

Closed square lattice (torus) is spherical

Closed graphene lattice is spherical

Sodalite closed zeolitic lattice

is spherical =1 too and so onand so on..


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Assuming to determine sphericity produces

interesting (and sometime unexpected) results:

C60 is spherical =1 as we said;

C20

polyhedron is also spherical =1

Other fullerenes Cn have >1 so less

spherical than C20but surely more stable

Warning: better we try correlating and

relative stability only among isomers of the

same Cn fullerene.


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is then the topological sphericityof a chemical graph

The lower is the more spherical

and stable the structure will be. So we adopted for invariant the

new alias topological sphericitytopological sphericity

indexindex


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The minimum principle works BETTER on topological efficiency indextopological efficiency index

1 as it has been successfully tested on C66 fullerene 4478 isomers, see

Topological efficiency of C66 fullerene, D. Vukicevic, F. Cataldo , O. Ori,A. Graovac, Chemical Physics Letters 501 (2011) 442445.


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Int. J. Mol. Sci. 2011, 12(11), 7934-7949;

doi:10.3390/ijms12117934Topological Anisotropy of Stone-Wales Waves in Graphenic

FragmentsOttorino Ori, Franco Cataldo and Mihai V. Putz


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Index has been put in FULLERENE generator

program by Prof. Peter Schwerdtfeger

http://ctcp.massey.ac.nz/index.php?group=&page

=fullerenes&menu=fullerenes

free DOWNLAOD PLS USE IT

Indices W andIndices W and tool quickly sieve interesting candidatestool quickly sieve interesting candidatesfor stable chemical structures (molecules, 1for stable chemical structures (molecules, 1--22--3D3D

lattices, fractalslattices, fractals))

WEST UNIVERSITY OF TIMIWEST UNIVERSITY OF TIMIOARAOARA


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Questions ?

Ottorino ORIActinium Chemical Research

Rome

[email protected]

WEST UNIVERSITY OF TIMIWEST UNIVERSITY OF TIMIOARAOARAFACULTY OF CHEMISTRY, BIOLOGY, GEOGRAPHY

DEPARTMENT OF BIOLOGYDEPARTMENT OF BIOLOGY--CHEMISTRYCHEMISTRYLAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICALLAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICAL--CHEMISTRY FORCHEMISTRY FOR

NANOSCIENCES & QSARNANOSCIENCES & QSAR(CFSCNQ)(CFSCNQ)

Prof. dr. dr. habil. Mihai V. PUTZProf. dr. dr. habil. Mihai V. PUTZ

Director CFSCNQDirector CFSCNQ

[email protected][email protected]

topo chemistry nano chemistry lecture 01

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